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Gianmarco Caldini

Publications and source records attributed to Gianmarco Caldini.

7 recordsLinked to original sources
arXiv2025

On smooth approximation of integral cycles mod 2

We prove that every mod 2 integral cycle $T$ in a Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by a cycle which is a smooth submanifold $Σ$ of nearly the same area, up to a singular set of codimension 3; in addition, this estimate on the singular set can be refined depending on the codimension of the cycle. Moreover, if the mod 2 homology class $τ$ admits a smooth embedded representative, then $Σ$ can be chosen free of singularities. This article provides the unoriented version of the smooth approximation theorem for integral cycles.

math.DG↗
arXiv2025

Hausdorff measure bounds for density-$Q$ flat singularities of minimizing integral currents

In this article we prove that the set of flat singular points of locally highest density of area-minimizing integral currents of dimension $m$ and general codimension in a smooth Riemannian manifold $Σ$ has locally finite $(m-2)$-dimensional Hausdorff measure. In fact, the set of such flat singular points can be split into a union of two sets, one of which we show is locally $\mathcal{H}^{m-2}$-negligible, while for the other we obtain local $(m-2)$-dimensional Minkowski content bounds.

math.DG↗
arXiv2024

Optimal smooth approximation of integral cycles

In this article we prove that each integral cycle $T$ in an oriented Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by an integral cycle in the same homology class which is a smooth submanifold $Σ$ of nearly the same area, up to a singular set of codimension 5. Moreover, if the homology class $τ$ is representable by a smooth submanifold, then $Σ$ can be chosen free of singularities.

math.DG↗
arXiv2023

Generic uniqueness for the Plateau problem

Given a complete Riemannian manifold $\mathcal{M}\subset\mathbb{R}^d$ which is a Lipschitz neighbourhood retract of dimension $m+n$, of class $C^{h,β}$ and an oriented, closed submanifold $Γ\subset \mathcal M$ of dimension $m-1$, which is a boundary in integral homology, we construct a complete metric space $\mathcal{B}$ of $C^{h,α}$-perturbations of $Γ$ inside $\mathcal{M}$, with $α<β$, enjoying the following property. For the typical element $b\in\mathcal B$, in the sense of Baire categories, there exists a unique $m$-dimensional integral current in $\mathcal{M}$ which solves the corresponding Plateau problem and it has multiplicity one.

math.AP↗
arXiv2022

Well-posedness properties of geometric variational problems: existence, regularity and uniqueness results

This thesis is devoted to the study of well-posedness properties of some geometric variational problems: existence, regularity and uniqueness of solutions. We study two specific problems arising in the context of geometric calculus of variations and sharing strong analogies: the Plateau's problem and the optimal branched transport problem. The first part of the thesis discusses the existence theory. Both problems are formulated in the language of Federer and Fleming's theory of currents. After an exposition of the main results, we will present the core ideas of the (interior) regularity theory for area-minimizing currents and for optimal transport paths. The last part of the thesis contains two original results: the generic uniqueness of solutions both for the Plateau's problem (in any dimension and codimension) and for the optimal branched transport problem.

math.DG↗
arXiv2022

On the notion of measurable utility on a connected and separable topological space: an order isomorphism theorem

The aim of this article is to define a notion of cardinal utility function called measurable utility and to define it on a connected and separable subset of a weakly ordered topological space. The definition is equivalent to the ones given by Frisch in 1926 and by Shapley in 1975 and postulates axioms on a set of alternatives that allow both to ordinally rank alternatives and to compare their utility differences. After a brief review of the philosophy of utilitarianism and the history of utility theory, the paper introduces the mathematical framework to represent intensity comparisons of utility and proves a list of topological lemmas that will be used in the main result. Finally, the article states and proves a representation theorem for a measurable utility function defined on a connected and separable subset of a weakly ordered topological space equipped with another weak order on its cartesian product. Under some assumptions on the order relations, the main theorem proves existence and uniqueness, up to positive affine transformations, of an order isomorphism with the real line.

econ.GN↗